Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(307,666)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("666.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,6,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 307.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 666.307
Dual form 666.2.s.e.397.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.50000 + 0.866025i) q^{5} +(-2.00000 + 3.46410i) q^{7} +1.00000i q^{8} -1.73205 q^{10} -4.73205 q^{11} +(-5.19615 - 3.00000i) q^{13} -4.00000i q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.50000 - 0.866025i) q^{17} +(-1.09808 - 0.633975i) q^{19} +(1.50000 - 0.866025i) q^{20} +(4.09808 - 2.36603i) q^{22} -1.26795i q^{23} +(-1.00000 - 1.73205i) q^{25} +6.00000 q^{26} +(2.00000 + 3.46410i) q^{28} +4.26795i q^{29} -1.26795i q^{31} +(0.866025 + 0.500000i) q^{32} +(-0.866025 + 1.50000i) q^{34} +(-6.00000 + 3.46410i) q^{35} +(-0.500000 - 6.06218i) q^{37} +1.26795 q^{38} +(-0.866025 + 1.50000i) q^{40} +(0.232051 - 0.401924i) q^{41} +9.46410i q^{43} +(-2.36603 + 4.09808i) q^{44} +(0.633975 + 1.09808i) q^{46} -11.6603 q^{47} +(-4.50000 - 7.79423i) q^{49} +(1.73205 + 1.00000i) q^{50} +(-5.19615 + 3.00000i) q^{52} +(-1.26795 - 2.19615i) q^{53} +(-7.09808 - 4.09808i) q^{55} +(-3.46410 - 2.00000i) q^{56} +(-2.13397 - 3.69615i) q^{58} +(2.19615 - 1.26795i) q^{59} +(12.6962 + 7.33013i) q^{61} +(0.633975 + 1.09808i) q^{62} -1.00000 q^{64} +(-5.19615 - 9.00000i) q^{65} +(-3.09808 + 5.36603i) q^{67} -1.73205i q^{68} +(3.46410 - 6.00000i) q^{70} +(1.26795 - 2.19615i) q^{71} -12.3923 q^{73} +(3.46410 + 5.00000i) q^{74} +(-1.09808 + 0.633975i) q^{76} +(9.46410 - 16.3923i) q^{77} +(-7.09808 - 4.09808i) q^{79} -1.73205i q^{80} +0.464102i q^{82} +(5.83013 + 10.0981i) q^{83} +3.00000 q^{85} +(-4.73205 - 8.19615i) q^{86} -4.73205i q^{88} +(-4.50000 + 2.59808i) q^{89} +(20.7846 - 12.0000i) q^{91} +(-1.09808 - 0.633975i) q^{92} +(10.0981 - 5.83013i) q^{94} +(-1.09808 - 1.90192i) q^{95} +5.19615i q^{97} +(7.79423 + 4.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 6 q^{5} - 8 q^{7} - 12 q^{11} - 2 q^{16} + 6 q^{17} + 6 q^{19} + 6 q^{20} + 6 q^{22} - 4 q^{25} + 24 q^{26} + 8 q^{28} - 24 q^{35} - 2 q^{37} + 12 q^{38} - 6 q^{41} - 6 q^{44} + 6 q^{46}+ \cdots + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 1.50000 + 0.866025i 0.670820 + 0.387298i 0.796387 − 0.604787i \(-0.206742\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) −2.00000 + 3.46410i −0.755929 + 1.30931i 0.188982 + 0.981981i \(0.439481\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −1.73205 −0.547723
\(11\) −4.73205 −1.42677 −0.713384 − 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) 0 0
\(13\) −5.19615 − 3.00000i −1.44115 − 0.832050i −0.443227 − 0.896410i \(-0.646166\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) − 4.00000i − 1.06904i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 1.50000 − 0.866025i 0.363803 − 0.210042i −0.306944 − 0.951727i \(-0.599307\pi\)
0.670748 + 0.741685i \(0.265973\pi\)
\(18\) 0 0
\(19\) −1.09808 − 0.633975i −0.251916 − 0.145444i 0.368725 − 0.929538i \(-0.379794\pi\)
−0.620641 + 0.784095i \(0.713128\pi\)
\(20\) 1.50000 − 0.866025i 0.335410 − 0.193649i
\(21\) 0 0
\(22\) 4.09808 − 2.36603i 0.873713 − 0.504438i
\(23\) − 1.26795i − 0.264386i −0.991224 − 0.132193i \(-0.957798\pi\)
0.991224 − 0.132193i \(-0.0422018\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 2.00000 + 3.46410i 0.377964 + 0.654654i
\(29\) 4.26795i 0.792538i 0.918134 + 0.396269i \(0.129695\pi\)
−0.918134 + 0.396269i \(0.870305\pi\)
\(30\) 0 0
\(31\) − 1.26795i − 0.227730i −0.993496 − 0.113865i \(-0.963677\pi\)
0.993496 − 0.113865i \(-0.0363232\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) −0.866025 + 1.50000i −0.148522 + 0.257248i
\(35\) −6.00000 + 3.46410i −1.01419 + 0.585540i
\(36\) 0 0
\(37\) −0.500000 − 6.06218i −0.0821995 − 0.996616i
\(38\) 1.26795 0.205689
\(39\) 0 0
\(40\) −0.866025 + 1.50000i −0.136931 + 0.237171i
\(41\) 0.232051 − 0.401924i 0.0362402 − 0.0627700i −0.847336 − 0.531057i \(-0.821795\pi\)
0.883577 + 0.468287i \(0.155129\pi\)
\(42\) 0 0
\(43\) 9.46410i 1.44326i 0.692278 + 0.721631i \(0.256607\pi\)
−0.692278 + 0.721631i \(0.743393\pi\)
\(44\) −2.36603 + 4.09808i −0.356692 + 0.617808i
\(45\) 0 0
\(46\) 0.633975 + 1.09808i 0.0934745 + 0.161903i
\(47\) −11.6603 −1.70082 −0.850411 − 0.526118i \(-0.823647\pi\)
−0.850411 + 0.526118i \(0.823647\pi\)
\(48\) 0 0
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) 1.73205 + 1.00000i 0.244949 + 0.141421i
\(51\) 0 0
\(52\) −5.19615 + 3.00000i −0.720577 + 0.416025i
\(53\) −1.26795 − 2.19615i −0.174166 − 0.301665i 0.765706 − 0.643191i \(-0.222390\pi\)
−0.939872 + 0.341526i \(0.889056\pi\)
\(54\) 0 0
\(55\) −7.09808 − 4.09808i −0.957104 − 0.552584i
\(56\) −3.46410 − 2.00000i −0.462910 − 0.267261i
\(57\) 0 0
\(58\) −2.13397 − 3.69615i −0.280205 − 0.485329i
\(59\) 2.19615 − 1.26795i 0.285915 − 0.165073i −0.350183 − 0.936681i \(-0.613881\pi\)
0.636098 + 0.771608i \(0.280547\pi\)
\(60\) 0 0
\(61\) 12.6962 + 7.33013i 1.62558 + 0.938527i 0.985391 + 0.170305i \(0.0544754\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0.633975 + 1.09808i 0.0805149 + 0.139456i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −5.19615 − 9.00000i −0.644503 − 1.11631i
\(66\) 0 0
\(67\) −3.09808 + 5.36603i −0.378490 + 0.655564i −0.990843 − 0.135020i \(-0.956890\pi\)
0.612353 + 0.790585i \(0.290223\pi\)
\(68\) − 1.73205i − 0.210042i
\(69\) 0 0
\(70\) 3.46410 − 6.00000i 0.414039 − 0.717137i
\(71\) 1.26795 − 2.19615i 0.150478 − 0.260635i −0.780925 − 0.624624i \(-0.785252\pi\)
0.931403 + 0.363989i \(0.118585\pi\)
\(72\) 0 0
\(73\) −12.3923 −1.45041 −0.725205 − 0.688533i \(-0.758255\pi\)
−0.725205 + 0.688533i \(0.758255\pi\)
\(74\) 3.46410 + 5.00000i 0.402694 + 0.581238i
\(75\) 0 0
\(76\) −1.09808 + 0.633975i −0.125958 + 0.0727219i
\(77\) 9.46410 − 16.3923i 1.07853 − 1.86808i
\(78\) 0 0
\(79\) −7.09808 − 4.09808i −0.798596 − 0.461070i 0.0443840 − 0.999015i \(-0.485867\pi\)
−0.842980 + 0.537945i \(0.819201\pi\)
\(80\) − 1.73205i − 0.193649i
\(81\) 0 0
\(82\) 0.464102i 0.0512514i
\(83\) 5.83013 + 10.0981i 0.639940 + 1.10841i 0.985446 + 0.169991i \(0.0543740\pi\)
−0.345506 + 0.938417i \(0.612293\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −4.73205 − 8.19615i −0.510270 − 0.883814i
\(87\) 0 0
\(88\) − 4.73205i − 0.504438i
\(89\) −4.50000 + 2.59808i −0.476999 + 0.275396i −0.719165 − 0.694839i \(-0.755475\pi\)
0.242166 + 0.970235i \(0.422142\pi\)
\(90\) 0 0
\(91\) 20.7846 − 12.0000i 2.17882 − 1.25794i
\(92\) −1.09808 − 0.633975i −0.114482 − 0.0660964i
\(93\) 0 0
\(94\) 10.0981 − 5.83013i 1.04154 − 0.601332i
\(95\) −1.09808 − 1.90192i −0.112660 − 0.195133i
\(96\) 0 0
\(97\) 5.19615i 0.527589i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 7.79423 + 4.50000i 0.787336 + 0.454569i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 666.2.s.e.307.1 4
3.2 odd 2 74.2.e.a.11.2 ✓ 4
12.11 even 2 592.2.w.d.529.1 4
37.27 even 6 inner 666.2.s.e.397.1 4
111.8 even 12 2738.2.a.j.1.2 2
111.29 even 12 2738.2.a.f.1.2 2
111.101 odd 6 74.2.e.a.27.2 yes 4
444.323 even 6 592.2.w.d.545.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.a.11.2 ✓ 4 3.2 odd 2
74.2.e.a.27.2 yes 4 111.101 odd 6
592.2.w.d.529.1 4 12.11 even 2
592.2.w.d.545.1 4 444.323 even 6
666.2.s.e.307.1 4 1.1 even 1 trivial
666.2.s.e.397.1 4 37.27 even 6 inner
2738.2.a.f.1.2 2 111.29 even 12
2738.2.a.j.1.2 2 111.8 even 12